Jozsef Garai

Heat capacities of elastic solids

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Abstract

The work function is embedded in the equation describing the relationship between the constant volume and constant pressure heat capacities. The modification of the work function results that the relationship between these quantities must be changed accordingly. Using the newly derived work functions of elastic solids the description of the heat capacities and the relationship between the heat capacities are given for solid phase.

CONTENTS

1. Introduction

2. Heat capacity of gas phase

3. Heat capacity of solid phase

4. Conclusions

1. Introduction

The molar heat capacity [c(φ)]γ represents the energy required to raise the temperature of one mol substance by one degree of Kelvin. The general description can be given as:

       (1)

where φ = g (gas), s (solid), l (liquid), and γ = p (pressure), V (volume) , n is the number of moles, T is the temperature, and q is the heat supplied to the system. The energy added to the system can be utilized as thermal energy or mechanical work. The differential of the heat supplied to the system comprises both the thermal and work related energies. The thermodynamic conditions affects only the work related heat while the thermal related part remains unchanged regardless of the conditions.

[δq(φ)]γ = δq(φ)thermal + [δq(φ)work]γ       (2)

The heat capacity in the same manner can be divided into thermal and work related parts. The thermal related part will be called molar thermal heat capacity c(φ)thermal

       (3)

while the work related part will be called to molar work heat capacity [c(φ)work]γ

       (4)

The heat capacity is then the sum of the thermal and work related heat capacities.

[c(φ)]γ = c(φ)thermal + [c(φ)work]γ       (5)

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2. Heat capacities of gas phase

In gas phase the molar heat capacity at constant volume is

       (6)

At constant volume no mechanical work is done and the molar work heat capacity is zero.

[c(g)work]V = 0       (7)

The energy supplied to the system is entirely utilized as thermal energy.

       (8)

According to the first law of thermodynamic the differential of the internal energy of a system [U] is equal with the sum of the differentials of the heat and work supplied to the system.

dU = δq = δw = ncVdT - pdV       (9)

In gas phase at constant volume equation (9) reduces to

dU(g) = δq(g)V = ncVdT       (10)

At constant pressure the differential of the internal energy is :

dU (g) = [δq(g)]p + [δw(g)]p      (11)

Combining equation (10) with (11) gives the differential of the heat supplied to the system at constant pressure.

[δ(g)]p = [δq(g)]V - [δw(g)]p       (12)

The heat capacity at constant pressure is then

       (13)

The first term in equation (13) is the molar heat capacity at constant volume [Eq. (8)] while the second term is the molar work heat capacity at constant pressure

       (14)

The relationship between the constant pressure and volume heat capacities in gas phase can be written as :

[c(g)]p = [c(g)]V + [c(g)work]p       (15)

Using the equation of state for ideal gasses

pV = nRT ⇒ p(dV)p = nRT       (16)

and substituting this result into equation (14) gives the molar work heat capacity for gasses

       (17)

where R is the universal gas constant, NA is the Avogadro number, and kB is the Boltzmann constant. The relationship between the molar heat capacities at constant pressure and volume is

[c(g0]p = [c(g)]V + R       (18)

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3. Heat capacities of solid phase

The current consensus in solid thermodynamics assumes that equation (8) is valid regardless of the phase; therefore, it can be used in the same form for solids [1].

[c(s)]V = c(s)thermal ≈ c(s)Debye       (19)

where c(s)Debye is the theoretical value of the thermal heat capacity calculated by using the Debye concept. The relationship between the heat capacities is described [2] as :

       (20)

where αV is the volume coefficient of thermal expansion and KT is the bulk modulus. In this study the thermodynamic description of the heat capacities and the relationship between the heat capacities will be investigated in detail in solid phase.

Investigating solids in a state of equilibrium it has been demonstrated that correlation between the temperature and the pressure can exist only at constant volume [3]. This correlation is irreversible and works in the temperature pressure direction. The limited communication between the temperature and the pressure put constraints on the conversion of the thermal and the mechanical energies. In solid phase the pressure correlates to the elastic related volume change while the temperature to the thermal related volume change. These volume changes are not compatible with the exception of constant volume condition when the expanded volume converts completely to compressed volume. Based on theoretical consideration it had been suggested that the thermal related volume change do not result mechanical work. Separating the thermal and elastic related volume changes the work functions for each of the thermodynamic conditions have been derived [4].

Since only the elastic volume change results work in solid phase the molar work heat capacity should be written as :

       (21)

In solid phase contrarily to gasses the heat added to system does work on the system. The signs given in Eq. (14) for the differential of the work and for the work function have been changed in Eq. (21) accordingly. The work functions of solids [4] for constant pressure and volume are

       (22)

       (23)

Subscripts i and f are used for initial and final conditions respectively. Assuming that Ti = T and that Tf =T + 1K the internal work related to one degree of temperature change can be calculated

       (24)

and

       (25)

The molar work heat capacities for solids are

       (26)

and

       (27)

Substituting the molar volume at zero pressure .

       (28)

the heat capacities are at constant pressure [c(s)]p

       (29)

and at constant volume [c(s)]V

       (30)

Combining equation (29) and (30) the relationship between the constant pressure and volume heat capacities is

        (31)

Simplifying equation (31), the relationship between constant pressure and constant volume molar heat capacities can be rewritten like:

       (32)

where

       (33)

Assuming that αV x1o Kelvin is small compared to 1 then equation (32) can be written as :

       (34)

It can be seen from equations (26) and (27) that in solid phase the mechanical work related to one degree of temperature change at constant volume is greater than at constant pressure.

       (35)

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4. Conclusions

The higher value of the mechanical work at constant volume indicates that the value of the heat capacity at constant volume should be higher than at constant pressure.

[c(s)]V > [c(s)]p > [c(s)]thermal = [c(s)]p=0       (36)

The temperature and pressure dependence of the molar work heat capacities and their relationship to each other is shown on Fig. 1.

Using the conventional approach [Eq. (20)] and calculating the molar volume heat capacities from experiments the theoretical and experimental values did not show good agreement for different minerals5. The inconsistencies might be resulting from the incorrect theoretical description of the heat capacity.

Acknowledgement:

I would like to thank Alexandre Laugier for his encouragement and helpful comments on the manuscript.

References:

[1] The word of solid in this text is used for homogeneous, isotropic, linear elastic, non-viscous material.

[2] F. Reif, Fundamentals of statistical and thermal physics, McGraw-Hill Inc., International Edition, Singapore, 1985, p. 168

[3] J. Garai (2005) Limits on the Thermo-elastic Coupling of Solids, http://arxiv.org/physics/0507075

[4] J. Garai, and A. Laugier (2005) The work function of elastic solid, http://arxiv.org/physics/0504202

[5] S.W. Kieffer (1979) Thermodynamics and Lattice Vibrations of Minerals: 1. Mineral Heat Capacities and Their relationships to Simple Lattice Vibration Models; 2. Vibrational characteristics of Silicates; 3. Lattice Dynamics and Approximation for Minerals With Application to Simple Substances and Framework Silicates, Rev. Geophys. Space Phys. 17 1-59

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